We reformulate Shimura's theory of nearly holomorphic forms for Siegel modular forms using automorphic sheaves over Siegel varieties. This sheaf-theoretic reformulation allows us to define and study basic properties of nearly overconvergent Siegel modular forms as well as their p-adic families. Besides, it finds applications in the construction, via the doubling method, of p-adic partial standard L-functions associated to Siegel cuspidal Hecke eigensystems. We illustrate how the sheaf-theoretic definition of nearly holomorphic forms and Maass--Shimura differential operators helps with the choice of the archimedean sections for the Siegel Eisenstein series on the doubling group Sp(4n) and the study of the p-adic properties of their restrictions to Sp(2n)*Sp(2n). The selection of archimedean sections, together with p-adic interpolation considerations, then naturally gives the sections at the place p. We compute p-adic zeta integrals corresponding to those sections. Finally, we construct the p-adic standard L-functions associated to ordinary families of Siegel Hecke eigensystems and obtain their interpolation properties.